Question: Regular hexagon $ABCDEF$ is the base of right pyramid $\allowbreak PABCDEF$.  If $PAD$ is an equilateral triangle with side length 8, then what is the volume of the pyramid?
Solution: [asy]
import three;
triple A = (1,0,0);
triple B = (0.5,sqrt(3)/2,0);
triple C = (-0.5,sqrt(3)/2,0);
triple D = (-1,0,0);
triple EE = (-0.5,-sqrt(3)/2,0);
triple F = (0.5,-sqrt(3)/2,0);

triple P = (0,0,1);

draw(F--A--B--C);
draw(C--D--EE--F,dashed);
draw(A--P--C);
draw(EE--P--D,dashed);
draw(B--P--F);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,E);
label("$D$",D,S);
label("$P$",P,N);
label("$E$",EE,S);
draw(A--D,dashed);
label("$F$",F,W);
draw(EE--B,dashed);
draw(C--F,dashed);
triple O = (0,0,0);
draw(P--O,dashed);
label("$O$",O,S);
[/asy]

Drawing the long diagonals of a regular hexagon divides the hexagon into equilateral triangles with side length equal to half the length of each long diagonal.  So, the area of the base equals 6 times the area of an equilateral triangle with side length 4.  An equilateral triangle with side length 4 has area $4^2\sqrt{3}/4 = 4\sqrt{3}$, so the area of the base of the pyramid is $6(4\sqrt{3}) = 24\sqrt{3}$.

Let $O$ be the center of the hexagon, so $\overline{PO}$ is the altitude from the apex of the pyramid.  Since triangle $PAD$ is an equilateral triangle, triangle $POA$ is a 30-60-90 triangle with hypotenuse 8.  $\overline{PO}$ is opposite the $60^\circ$ angle in this triangle, so $PO = 4\sqrt{3}$.

Finally, the volume of the pyramid is  \[\frac13\cdot [ABCDEF] \cdot PO = \frac13\cdot 24\sqrt{3} \cdot 4\sqrt{3} = \boxed{96}.\]